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Bruce Stark said-

>The procedure I posted March 30, for calculating altitudes when both GMT and
>longitude are wildly uncertain, works fine.
>
>It's the same system navigators from James Cook to Joshua Slocum used to find
>local hour angles. The only difference is it uses sidereal hour angle instead
>of right ascension, so things are measured from east to west.
>
>I've found it's less confusing if I first take only the GHAs of Aries and the
>sun from the Almanac with the guessed-at GMT. That way, fewer numbers are
>lying around to stumble over. Only after finding SHA meridian do I take the
>other numbers from the Almanac, with the same guessed-at GMT of course.
>
>The acid test for the method was the lunar Chuck Griffiths posted March 21.
>Here both bodies are near prime vertical, so change in hour angle affects the
>altitudes nearly 100%. What makes it exceptional is that the moon, Venus, and
>the observer are all in nearly the same plane, so the distance is affected
>nearly 100% by refraction and parallax. Total correction in altitude for
>parallax and refraction was 45.'6, of which 44.'9 showed up in the difference
>between the apparent and cleared distances. It would be hard to find a
>situation where the distance could be more sensitive to changes in the GMT
>used to calculate the altitudes. It couldn't happen except in low latitudes.
>
>Since Chuck neglected to take an observation for local apparent time I did,
>in this case, have to know his longitude and accurate GMT to get his LAT. I
>did so by applying the equation of time to GMT to get GAT, and his longitude
>to GAT to get LAT. LAT was March 18th, 18:36:35. That's 6:36:35 past noon.
>
>I pretended Chuck had found his LAT by observation before the sun went down.
>I also pretended he'd so egregiously screwed up his dead reckoning that it
>put him seven and a half degrees, nearly 400 nautical miles at his latitude,
>east of where he was. This made his estimated GMT half an hour less than the
>true.
>
>The first try got a GMT of 00:20:19, March 19th.
>
>That's nearly half an hour from the GMT used to calculate the altitudes, so a
>repeat was called for.
>
>The second try got GMT of 00:18:33.
>
>Obviously nothing much to be gained by going further. Half an hour off in the
>GMT used to calculate the altitudes affected the result less than two
>minutes. Two minutes off can be ignored. This may ease George's concerns
>about the use of calculated altitudes for clearing distances.
>
>Bruce

=====================

Response from George Huxtable-

Thank you, Bruce, for putting together a procedure which minimises an
observer's dependence on his initial guess at position and GMT, when he
works a lunar using a calculated altitude.

I think I have mostly understood the procedure, but I need another day or
two of thrashing it around in my mind.

Now let me spell out the consequences, to see whether I have got them
right. I hope that Bruce, or anyone else, will point out errors in my
conclusions.

1. For a lunar that uses altitudes that are calculated rather than
measured, it is necessary to make a measurement for local apparent time.
This should be done within a day or so of the moment of taking the lunar,
depending on how well the timepiece can be trusted over such a period.

2. A measure of local apparent time can be made by equal-altitudes, looking
for the moment of symmetry between the Sun's rise and fall, and correcting
for the North or South motion of the vessel and the Sun. Or alternatively,
if the latitude has been measured by a recent meridian altitude, and
corrected since then by the motion of the ship, a single measure of the
Sun's altitude in the morning or evening can be used to determine the local
apparent time.

4. The local apparent time, so derived, and corrected by the equation of
time to give local mean time, can be checked against the ship's timepiece,
and can now be used with the Almanac to establish precise altitudes for
both the bodies used in a lunar observation, even though the GMT and the
longitude are both still uncertain.

5. The lunar observation can then be precisely corrected using these known
altitudes, to give a good value for GMT, and the error of the timepiece
against GMT can be found. If necessary, a reiteration can be made.

6. Because the timepiece has recently been checked against local mean time,
the time difference between local mean time and Greenwich mean time can be
found, and this gives directly the ship's longitude that we were seeking.

On the other hand, if we were able to actually measure the altitudes of the
two bodies used in a lunar, then we know that those altitudes must
correspond to the moment that the lunar was taken, whenever that happened
to be. So-

7. We can work the lunar using those measured altitudes. That will give the
GMT at which the lunar was taken, against which we can check the timepiece.

8. Using that GMT, we can look up the Dec and GHA of the two bodies at that
moment, and from the Almanac, predict their altitude and azimuth at the
observer's assumed position.

9. Comparing measured and predicted positions give an offset to draw on a
chart at right-angles to the azimuth, and providing that these cross at a
sensible angle-of-cut, their intersection provides the measured position in
lat and long.

Navigators will recognise steps 8 and 9 as standard position-line
navigation, with which they will be very familiar. This is the procedure
they can adopt if altitudes in a lunar have been measured rather than
calculated. It's exactly the same as chronometer navigation using position
lines.

But if the altitudes have instead been calculated, then steps 1 to 6 must
be taken instead. The procedure then, requiring the finding of local time,
is what was used in the days before Sumner invented position lines in 1837.
In those early days, latitude and longitude were considered as quite
different quantities, to be determined independently. When the local time
has been obtained in this method, its difference from GMT gives the
longitude directly, and there is no requirement for any further
"position-line navigation".

The point I wish to bring out here is that in choosing between measuring
and calculating the altitudes, a very different approach to the necessary
observations and calculations will be required.

All these problems stem from the strong effects that changes of Moon
altitude, and hence Moon parallsx, have on the position of the Moon in the
sky.

George Huxtable.



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george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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